A conventional coplanar waveguide resonator is shown in FIG. 11. Hereinafter the coplanar waveguide resonator may be sometimes called as ‘resonator’.
Formed on a dielectric substrate 11 is a center conductor 12a, and a pair of ground conductors 13a and 13a′ are formed on the substrate 11 on the opposite sides of the center conductor 12a with a gap portion of a spacing ‘s’ therebetween where the dielectric 11 is exposed. At one end of the center conductor 12a, one side 212a thereof is connected in a short-circuit manner with the ground conductor 13a by a shorting end 14a while the other side 212a′ is connected in a short-circuit manner with the ground conductor 13a′ by a shorting end 14a′. The other ends of the ground conductors 13a and 13a′ are connected together by a ground conductor connector 13con, and the other end of the center conductor 12a is disposed opposite to the ground conductor connector 13#con with a spacing g therebetween. While the shorting ends 14a and 14a′ and the ground conductor connector 13con are shown as delineated by dotted lines, they are formed integrally with the ground conductors and the center conductor by appearance. The combination of the center conductor 12a, the ground conductors 13a and 13a′ and the shorting ends 14a and 14a′ defines a coplanar line having a characteristic impedance which is determined by a ratio of the width w of the center conductor 12a to the distance w+2s between the ground conductors 13a and 13a′. Since the center conductor 12a and the ground conductors 13a and 13a′ are formed to be coplanar, it is a simple matter to form the shorting ends 14a and 14a′. In other words, a microwave circuit using a coplanar line has a greater freedom of design and is more readily manufactured as compared with a microwave circuit using a microstrip line which requires via-holes.
In one example of the coplanar line, the dielectric substrate 11 has a dielectric constant of 9.68. The substrate 11 has a thickness Lc=0.5 mm. The conductor is made of superconducting material and has a thickness Ld=0.5 μm, w=218 μm, and s=91 μm.
The center conductor 12a has a length L1 which is electrically equivalent to one-quarter wavelength, and accordingly, a resonance occurs with a high frequency signal which has such a wavelength. In the description to follow, the ground conductors 13a and 13a′ may be generically referred to as a ground conductor 13, and the shorting ends 14a and 14a′ may be generically referred to as a shorting end 14, which is also referred to as a stub.
A plurality of coplanar waveguide resonators may be connected in a cascade connection to form a coplanar filter, as disclosed in a non-patent literature 1: T. TSUJIGUCHI et al. “A Miniaturized End-Coupled Bandpass Filter using λ/4 Hair-pin Coplanar Resonators”, p. 829, 1998 IEEE MTT-S Digest; a non-patent literature 2: I. AWAI et al. “Coplanar Stepped-Impedance-Resonator Bandpass Filter”, pp. 1–4, 2000 China Japan Joint Meeting On Microwaves; and a non-patent literature 3: H. SUZUKI et al. “A Low-Loss 5 GHz Bandpass Filter Using HTS Quarter-Wavelength Coplanar Waveguide Resonators”, pp. 714–719, IEICE TRANS.ELECTRON., VOL. E85-C, NO.3 March 2002.
An example of the coplanar filter constructed with coplanar waveguide resonators of FIG. 11 is shown in FIG. 12A. In this example, four coplanar waveguide resonators 15a, 15b, 15c and 15d are formed on a common dielectric substrate 11 and are in cascade connection. The resonators 15a and 15b share the shorting end 14 in common. Specifically, two ground conductors 13a and 13a′, two shorting ends 14a and 14a′ and the center conductor 12a of the resonator 15a are in common with two ground conductors 13a and 13a′, two shorting ends 14b and 14b′ and center conductor 12b of the resonator 15b, forming a so-called foot-to-foot arrangement (inductive coupler) 16ab to couple the both resonators. The resonators 15b and 15c have their open edges of the center conductors 12b and 12c which are located far from the shorting ends 14b and 14c, and disposed close and opposite to each other, forming a top-to-top arrangement (capacitive coupler) 17bc to couple the both resonators. The resonators 15c and 15d share ground conductors 13c, 13c′; and 13d, 13d′; shorting ends 14c, 14c′; and 14d, 14d′; center conductors 12c and 12d in common, respectively to form the foot-to-foot arrangements (inductive coupler) 16cd which couples the both resonators. Thus, the capacitive coupling and the inductive coupling are used in alternate fashion to construct a filter having a bandpass response with four stage resonators. A coplanar line type input section 18 which is coupled to the open end of the resonator 15a which is disposed at one end of the cascade connection by a capacitive coupler 17ia and a coplanar line type output section 19 which is coupled to the open end of the resonator 15d disposed at the other end by a capacitive coupler 17do are formed on the dielectric substrate 11 sharing the ground conductors 13 in common. The capacitive couplers 17ia and 17do which couple between the input section 18 and the output section 19 on one hand and the resonators 15a and 15d on the other hand have a greater degree of coupling than the capacitive coupler 17bc disposed between the resonators 15b and 15c. 
The current density distribution of the filter shown in FIG. 12A which is calculated according to the electromagnetic field simulation using the moment method is shown in FIG. 13. The calculation has been made under the following conditions:
itemconditioninput signalsine wave of voltage 1 Vppport termination50 Ωfrequency 5 GHz
In this calculation, a simulation is made using coordinate axes shown as X-Y in FIG. 12A. Accordingly, in FIG. 13, a position on the X-axis indicated by X0 corresponds to the input end of the input section 18, and a position indicated by X6 corresponds to the output end of the output section 19. Each of positions X1 to X5 corresponds to the capacitive coupler 17ia, the inductive coupler 16ab, the capacitive coupler 17bc, the inductive coupler 16cd and the capacitive coupler 17do, respectively.
In each of the resonators 15a to 15d, the current density distribution is generally sinusoidal having a node at the open end and an antinode at the shorting end 14. It is seen that peaks in the current density distribution occurs at the coupler 16ab between the resonators 15a and 15b and the coupler 16cd between the resonator 15c and 15d, namely at locations where the sinusoidal current density distribution has maxima. This is because a current concentration occurs at the respective edge lines, namely the edge line 112a (see FIG. 12B) of intersections between the lateral side surface and the top surface of the center conductor 12a, the edge line 113a (see FIG. 11) between the lateral side surface 13a0 and the top surface of the ground conductor 13a and the edge line 20a between the lateral side surface 14a0 (see FIG. 11) and the top surface of the shorting end 14a, which is a so-called edge effect, and also because a current concentration further occurs at the corner area 21a1 and 21a2 (indicated as encircled by dotted lines in FIGS. 12A and 12B) since they have an angle of 90° formed between the edge line 20a which is viewed as a straight line in plan view of FIGS. 12A and 12B of the shorting end 14a and the edge line 112a of the center conductor 12a or the edge line 113a of the ground conductor 13a which is also viewed in the plan view.
The shorting end 14a which shorts the center conductor 12a to the ground conductor 13a is defined here to have the edge line 20 of a rectilinear configuration toward the dielectric. As seen from FIG. 11, the shorting end 14a has a lateral side surface 14a0 that have a height equal to the thickness of the conductor film by a length ‘s’ and a top surface. These surfaces intersect together with an edge line 20a therebetween. The lateral side surface 14a0 faces toward the gap portion of a spacing ‘s’ formed between the center conductor 12a and the ground conductor 13a where the dielectric 11 is exposed. The edge line 20a is seen as a straight line viewed in a plan view of FIG. 12B, thus it is defined the edge line toward the dielectric of the shorting edge 14a. Other edge line 112a of the center conductor 12a and still other edge line 113a of the ground conductor 13a are also seen straight lines in the plan view, thus they are fined in the same manner as being toward the dielectric. Any edge line other than those mentioned above is defined in the same manner as being toward the dielectric.
In order to consider the operation of the coupler 16ab, a combination of the two resonators 15a and 15b as shown in FIG. 12B (driver is not shown) is taken out from the filter shown in FIG. 12A. An exemplary current density distribution at one shorting end 14a of one resonator 15a is determined by a simulation as mentioned above on the basis of the construction shown in FIG. 12B in which a connecting portion 13con is provided between the ground conductors 13a and 13a, and a result of the simulation is shown in FIG. 14.
In FIG. 14, the calculation is based on the coordinate axes indicated by x-y axes as shown in FIG. 12B. Position yA on the y-axis corresponds to the position of a straight line 113a which represents an edge line toward the dielectric 11 of the ground conductor 13a, and position yB corresponds to the position of a straight line 112a which represents an edge line toward the dielectric of the center conductor 12a of the resonator 15a. Position xA on the x-axis corresponds to the position of a straight line 20a which represents an edge line toward the dielectric of the shorting end 14a. 
It will be evident from FIG. 14 that sharp peaks occur in the current density distribution at the respective corner points (bends) of the corner areas 21a1 and 21a2 and a maximum current density of 1365.5 A/m occurs at the corner point 121a2 of the corner area 21a2 where the shorting end 14a and the center conductor 12a are connected. It is to be noted that the current density distribution at the corner points of two other corner areas 21a2′, 21a1′ of the other shorting end 14a′(only indicated as encircled by dotted lines) is omitted from illustration in FIG. 14. The origin for the x axis and the y axis is as shown in FIG. 12B.
It is to be noted while a corner area has been generically referred to as 21 in the above description, postfix letters are used in FIG. 12A in order to identify a particular corner area. The same principle applies in the description to follow when a particular one is specifically identified.
The corner area 21a1 is formed by the intersection of the straight line 20a which represents an edge line toward the dielectric of the shorting end 14a and a straight line 113a which represents an edge line toward the dielectric of the ground conductor 13a of the resonator 15a at the corner point 121a1, and has an angle θ1 formed between the both straight lines, and the angle θ1 is 90° toward the dielectric. The corner area 21a2 is formed by the intersection of the edge line 20a toward the dielectric of the shorting end 14a and a straight line 112a which represents an edge line toward the dielectric of the center conductor 12a at the corner point 121a2, and has an angle θ2 formed between the both straight lines, and the angle θ2 is 90° toward the dielectric. Similarly, the other shorting end 14a′ which shorts the center conductor 12a and the ground conductor 13a′ of the resonator 15a has an edge line which forms an angle θ2′ of 90° toward the dielectric with the edge line toward the dielectric of the center conductor 12a and an angle θ1′ of 90° toward the dielectric with the edge line toward the dielectric of the ground conductor 13a′. 
It is stipulated here that an angle of such a corner area which is referred to hereafter refers to an angle toward the dielectric which is exposed at the gap portion.
In a conventional coplanar resonator, because the corner area of the shorting end has an angle of 90°, a sharp peak occurs at the corner points of the shorting end 14 where the current density distribution has its maximum, and this has been a cause of an increased power loss.
In the coplanar resonator in which the conductor is formed of a superconducting material, there is a critical current level which is inherent to the superconducting material, and even though the resonator were cooled to a temperature below a critical temperature, the superconducting state will be destroyed if a current which exceeds a critical current density flows through a portion thereof.